On triple coverings of irrational curves (Q1375758)

We investigate the problem of the existence of base-point-free pencils of relatively low degree on a triple covering \(X\) of genus \(g\) of a general curve \(C \) of genus \(h>0\). Such a problem is classical and the picture is rather well known for the degree range close to the genus \(g\). On the other hand, by a simple application of the Castelnuovo-Severi inequality one can easily see that there does not exist a base-point-free pencil of degree less than or equal to \((g-3h)/2\) other than the pull-backs from the base curve \(C\); while for the degree beyond this range not many things have been known about the existence of such a pencil which is not composed with the given triple covering. The main result of this paper is the following: Theorem A. Let \(X\) be a smooth algebraic curve of genus \(g\), over an algebraically closed field \(k\) with \(\text{char} (k)=0\), which admits a three sheeted covering onto a general curve \(C\) of genus \(h\geq 1\), \(g\geq (2[(3h+1)/2] +1) ([(3h+1)/2] +1)\). Then there exists a base-point-free pencil of any degree \(d\geq g- [(3h+1)/2]-1\) which is not composed with the given triple covering.